We study a natural LASSO variant of the
Multivariate Adaptive Regression Splines (MARS) method (see
[1]} or
[2]}) for nonparametric
regression. To understand the relation between a response
variable \(y\)  and \(d\)  explanatory variables \(x_1, \dots , x_d\)  based on
observed data \((x^{(1)}, y_1), \dots , (x^{(n)}, y_n)\)  with \(x^{(i)}\in \mathbb {R}^d\)  and \(y_i \in \mathbb {R}\) , MARS
fits a function \(y = \hat{f}_{\text{mars}}(x_1, \dots , x_d)\)  where
\(\hat{f}_{\text{mars}}\)  is a sparse linear combination of the functions of the
form:
\(\prod _{j=1}^d (b_j(x_j))^{\alpha _j} = \prod _{j: \alpha _j=1} b_j(x_j)\) 
